Proposition 22

Let A be a ring. Every quotient module of $A_s$ is monogenous. Conversely, let E be a monogenous A-module, c a generator of E and a its annihilator; the linear mapping $\xi \mapsto \xi c$ defines, when passing to the quotient, an isomorphism of $A_s/a$ onto E.

As $A_s$ is itself monogenous, being generated by 1, the first assertion follows from no. 7, Corollary 1 to Proposition 9. The second is obvious, since $\xi \mapsto \xi c$ is by hypothesis surjective and has kernel a.

Note that, if A is not commutative, the annihilators of two distinct generators c, c' of a monogenous A-module E are in general distinct and are also distinct from the annihilator of the module E. On the other hand, if A is commutative, the annihilator of a generator c of E is contained in the annihilator of every element of E and hence is the annihilator of the whole of E.